Attaining Quantum Sensing Enhancement from Monitored Dissipative Time Crystals

TL;DR

This work establishes that continuously monitored open quantum systems with dissipative time-crystal dynamics can achieve quantum-enhanced parameter estimation, with a universal scaling of the global QFI rate $f_{global} \sim N^3$ in the time-crystal phase for BTCs. It shows that a broad class of dissipative time-crystal models, including BTC and generalized dephasing systems, share this enhancement via closing Liouvillian gaps rather than relying on dissipative phase transitions, and derives an upper bound that separates generator variance from dissipative spectral properties. Crucially, the authors demonstrate that ideal monitoring ($\eta=1$) saturates the bound for both continuous homodyne and photodetection in BTC and a transverse collective dephasing (TCD) model, while inefficiencies degrade the BTC scaling to SQL, though the TCD model remains robust with efficiency-dependent super-classical scaling. The results identify monitored dissipative time crystals as a practical and robust platform for quantum metrology, with TCD offering particularly favorable resilience to measurement imperfections and a clear path to experimental realization.

Abstract

This study investigates quantum-enhanced parameter estimation through continuous monitoring in open quantum systems that exhibit a dissipative time crystal phase. We first analytically derive the global quantum Fisher information (QFI) rate for boundary time crystals (BTCs), demonstrating that within the time-crystal phase, the ultimate precision exhibits a cubic scaling with the system size, $f_{\mathrm{global}}\sim N^3$. We then generalize this finding to a broader class of dynamics, including the transverse collective dephasing (TCD) model, which achieves a time-crystal phase through a closing Liouvillian gap without requiring a dissipative phase transition. We proceed to numerically demonstrate that this maximal global QFI rate is experimentally attainable for both the BTC and TCD models, even at finite system sizes, via continuous homodyne and photodetection. Moving towards practical implementations, we analyze the precision limits under inefficient detection, revealing a critical difference: for BTC dynamics, inefficiencies asymptotically restore a classical scaling, and only a constant-factor quantum advantage remains possible. In contrast, for TCD dynamics, a super-classical scaling is still in principle observable, and our numerical simulations confirm its presence, even under inefficient measurement conditions, establishing the TCD model as a highly robust platform for quantum metrology.

Figures (4)

• Figure 1: The steady-state global QFI rate $f_{\text{global}}$ at $\omega = 4\kappa$ (blue circles) and $\omega = \kappa$ (orange circles). We demonstrate that the steady-state Fisher information rate from homodyne detection $f_{\text{signal}}$ is identical to $f_{\text{global}}$ at both $\omega = 4\kappa$ (red crosses) and $\omega = \kappa$ (black crosses). Additionally, we show that our analytic results for $f_{\text{global}}$ in the extreme time crystal limit in Eq. \ref{['eq:fglobal_analytic']} (green line) closely match the numerical results with $\omega = 4\kappa$.
• Figure 2: (a) The steady state Fisher information rate per spin of the measurement signal $f_{\text{signal}}/N$ for both homodyne (circles) and photodetection (triangles) in the BTC model. The rate is calculated for $\omega=4\kappa$ at two different measurement efficiencies $\eta = 0.9$ (red, top) and $\eta = 0.25$ (blue, bottom). The numerical results are compared against the theoretical bound from Eq. \ref{['eq:inefficiencybound']}. (b) The same as (a) for the TCD model. The rate is calculated for $\omega=0.1\kappa$. We leave the bound from (a) for reference.
• Figure S1: The scaling exponent $z_{k,N}$ of the three smallest eigenvalues with non-zero contribution to $f_{\text{global}}$ as a function of $N$ (circles). The equivalent scaling exponent of the imaginary parts of the eigenvalues is also shown when the eigenvalues have non-zero real part (triangles). The dashed lines show a power law scaling fit to $z_{k,N}$ for each eigenvalue. The black solid line shows $z = 1/3$, the value of the limit of all the power law scaling fits.
• Figure S2: The scaling exponent $\zeta_{N}$ of $f_{\text{global}}$ as a function of $N$ (circles). The black solid line is at $\zeta = 5/3$, a lower bound and proposed estimate of the value of $\zeta_N$ in the large N limit.